4.2 Kelvin Model
115
process has ended. The unloading phase activates again visco-elasticity and terminates with E (t 3 ) + η ˙
(t 3 ) = 0 with (t 3 ) = −˙ (t 3 ) ≈ 3 (from visual inspection).
The resulting σ = σ() diagram of nonlinear parallelogram format is highlighted
in Fig. 4.22c.
Finally, Fig. 4.22e, f depict the resulting σ = σ() diagrams for 10 and 100 times
smaller t 1 , t 2 , t 3 corresponding to higher stress rates | ˙
σ(t)|, respectively. They clearly
demonstrate a rigid behaviour with vanishing strain for | ˙
σ(t)| → ∞.
4.2.4 Generic Kelvin Model: Formulation
A generic formulation of the Kelvin model can be obtained from generalizing the
specific Kelvin model in Fig. 4.12 by assuming the elastic spring or/and the viscous
dashpot as nonlinear.
For the generic Kelvin model the free energy density ψ is expressed as a nonquadratic, yet convex, function of (the total strain)
ψ = ψ().
(4.76)
Then the energetic stress σ
follows as
σ
() = ∂ ψ().
(4.77)
Recall that the energetic and the dissipative stresses are constitutively related to
the total stress σ (that enters the equilibrium condition) by σ = σ
+ σ
. Moreover
the definitions σ e := σ
(in the elastic spring) and σ v := σ
(in the viscous dashpot)
are introduced.
Furthermore, for the generic Kelvin model it is possible to introduce the convex
and smooth (non-quadratic) dissipation and dual dissipation potentials as π = π(˙ )
and π
∗
= π
∗
(σ v ), respectively, which are related via corresponding Legendre transformations
π ( ˙
) = max
σ v
{σ v ˙
− π
∗
(σ v )},
(4.78a)
π
∗
(σ v ) = max
˙
{σ v ˙
− π ( ˙
)}.
(4.78b)
The stationarity conditions corresponding to Eqs. 4.78a and 4.78b are the constitutive relations
˙
(σ v ) = ∂ σ v π
∗
(σ v ),
(4.79a)
σ v ( ˙
) = ∂ ˙
π ( ˙
).
(4.79b)
Obviously, the relations in Eqs. 4.79a and 4.79b determine entirely the dissipative
behavior of the generic Kelvin model, thus the formulation is completed at this stage.
115
process has ended. The unloading phase activates again visco-elasticity and terminates with E (t 3 ) + η ˙
(t 3 ) = 0 with (t 3 ) = −˙ (t 3 ) ≈ 3 (from visual inspection).
The resulting σ = σ() diagram of nonlinear parallelogram format is highlighted
in Fig. 4.22c.
Finally, Fig. 4.22e, f depict the resulting σ = σ() diagrams for 10 and 100 times
smaller t 1 , t 2 , t 3 corresponding to higher stress rates | ˙
σ(t)|, respectively. They clearly
demonstrate a rigid behaviour with vanishing strain for | ˙
σ(t)| → ∞.
4.2.4 Generic Kelvin Model: Formulation
A generic formulation of the Kelvin model can be obtained from generalizing the
specific Kelvin model in Fig. 4.12 by assuming the elastic spring or/and the viscous
dashpot as nonlinear.
For the generic Kelvin model the free energy density ψ is expressed as a nonquadratic, yet convex, function of (the total strain)
ψ = ψ().
(4.76)
Then the energetic stress σ
follows as
σ
() = ∂ ψ().
(4.77)
Recall that the energetic and the dissipative stresses are constitutively related to
the total stress σ (that enters the equilibrium condition) by σ = σ
+ σ
. Moreover
the definitions σ e := σ
(in the elastic spring) and σ v := σ
(in the viscous dashpot)
are introduced.
Furthermore, for the generic Kelvin model it is possible to introduce the convex
and smooth (non-quadratic) dissipation and dual dissipation potentials as π = π(˙ )
and π
∗
= π
∗
(σ v ), respectively, which are related via corresponding Legendre transformations
π ( ˙
) = max
σ v
{σ v ˙
− π
∗
(σ v )},
(4.78a)
π
∗
(σ v ) = max
˙
{σ v ˙
− π ( ˙
)}.
(4.78b)
The stationarity conditions corresponding to Eqs. 4.78a and 4.78b are the constitutive relations
˙
(σ v ) = ∂ σ v π
∗
(σ v ),
(4.79a)
σ v ( ˙
) = ∂ ˙
π ( ˙
).
(4.79b)
Obviously, the relations in Eqs. 4.79a and 4.79b determine entirely the dissipative
behavior of the generic Kelvin model, thus the formulation is completed at this stage.